Onate 2004a - Revision history

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 In the FIC formulation the starting point are the modified differential equations of the problem  as previously described. These equations are however not useful to find an analytical solution, <math display="inline">\phi (x)</math>, for the physical problem. Nevertheless, the numerical solution of the FIC equation can be readily found. Moreover, by adequately choosing the values of the characteristic length parameter <math display="inline">h</math>, the numerical solution of the FIC equations will be always stable (physically sound) for any discretization level chosen.
-This process is schematically represented in Figure [[#img-1|1]] where it is shown that the numerical oscillations  for the coarser discretizations <math display="inline">M_1</math> and <math display="inline">M_2</math> dissapear when using the FIC procedure.
+This process is schematically represented in Figure [[#img-1|3]] where it is shown that the numerical oscillations  for the coarser discretizations <math display="inline">M_1</math> and <math display="inline">M_2</math> dissapear when using the FIC procedure.
 We can  conclude the FIC approach allows us to obtain a ''better numerical solution for a given discretization''. Indeed, as in the standard infinitesimal case, the choice of <math display="inline">M_\infty </math> will yield the (unreachable) exact analytical solution and this ensures the consistency  of the method.

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-<!-- metadata commented in wiki content
+Published in ''Finite Element Methods: 1970's and Beyond'', Franca L.P., Tezduyar T.E. and Masud A. (Eds.),CIMNE, Barcelona, Spain, 2004
-==STABILIZATION VIA FINITE CALCULUS IN FINITEELEMENT COMPUTATIONS==
-−
 '''
-Eugenio Oñate  '''
-−
-{|
-|-
-|   International Center for Numerical
-|-
-| Methods in Engineering (CIMNE)
-|-
-| Universidad Politecnica de Cataluna
-|-
-| Campus Norte UPC, 08034 Barcelona, Spain
-|-
-| Email: [mailto:onate@cimne.upc.es onate@cimne.upc.es]
-|-
-| Web page: [http://www.cimne.upc.es http://www.cimne.upc.es]
-|-
-| Dedicated to Tom Hughes on the occasion of his 60th birthday
-|}
--->
-−
 ==Abstract==

@@ Line 52: / Line 52: @@
 |}
-where [Galerkin] denotes the standard Galerkin expression, <math display="inline">{\boldsymbol P}(N_j)</math> is a vector which terms depend on the shape functions <math display="inline">N_j</math> (and the physical parameters of the problem), '''r''' is the vector of residuals of the finite element equations and <math display="inline">[\tau ^e]</math> is a matrix of stabilization parameters.
+where [Galerkin] denotes the standard Galerkin expression, <math display="inline">{\boldsymbol P}(N_j)</math> is a vector which terms depend on the shape functions <math display="inline">N_j</math> (and the physical parameters of the problem), <math>r</math> is the vector of residuals of the finite element equations and <math display="inline">[\tau ^e]</math> is a matrix of stabilization parameters.
 The computation of the stabilization parameters is still an open problem and much effort has been devoted to this topic <span id='citeF-1'></span>[[#cite-1|[1]],<span id='citeF-2'></span>[[#cite-2|2]],<span id='citeF-7'></span>[[#cite-7|7]]-<span id='citeF-21'></span>[[#cite-21|21]],<span id='citeF-36'></span>[[#cite-36|36]]].
-Different stabilized methods can be derived from Eq.([[#eq-1|1]]) by chosing expressions for matrices '''P''' and <math display="inline">[\tau ^e]</math>.  In this paper we will focus in the stabilized finite element formulation using the FIC method.  The equivalent form of Eq.([[#eq-1|1]]) is written in the FIC formulation as <span id='citeF-1'></span>[[#cite-1|[1]]-<span id='citeF-7'></span>[[#cite-7|7]]]
+Different stabilized methods can be derived from Eq.([[#eq-1|1]]) by chosing expressions for matrices <math>P</math> and <math display="inline">[\tau ^e]</math>.  In this paper we will focus in the stabilized finite element formulation using the FIC method.  The equivalent form of Eq.([[#eq-1|1]]) is written in the FIC formulation as <span id='citeF-1'></span>[[#cite-1|[1]]-<span id='citeF-7'></span>[[#cite-7|7]]]
 <span id='eq-2'></span>

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 ==9 CONCLUDING REMARKS==
-We have presented in this paper the possibilities of the finite calculus (FIC) method for deriving stabilized finite element formulations for a variety of problems in mechanics. In all cases the modified differential equations derived via the FIC method yield naturally a discretized system of equations with intrinsic stabilizations properties. These equations are more general than other standard stabilization methods (such as SUPG) and they allow to obtain correct numerical solutions for complex problems involving boundary layers and sharp internal layers. The key to the succes of the FIC method is the correct selection of the characteristic vector '''h'''. We have presented a new gradient-based definition of '''h''' which introduces naturally stabilization terms of laplacian-type in the equations for convective-diffusive transport and incompressible fluid flow problems.
+We have presented in this paper the possibilities of the finite calculus (FIC) method for deriving stabilized finite element formulations for a variety of problems in mechanics. In all cases the modified differential equations derived via the FIC method yield naturally a discretized system of equations with intrinsic stabilizations properties. These equations are more general than other standard stabilization methods (such as SUPG) and they allow to obtain correct numerical solutions for complex problems involving boundary layers and sharp internal layers. The key to the succes of the FIC method is the correct selection of the characteristic vector <math>h</math>. We have presented a new gradient-based definition of <math>h</math> which introduces naturally stabilization terms of laplacian-type in the equations for convective-diffusive transport and incompressible fluid flow problems.
-Numerical examples with applications of the formulation here presented can be found in <span id='citeF-43'></span>[[#cite-43|43]].
+Numerical examples with applications of the formulation here presented can be found in <span id='citeF-43'></span>[[#cite-43|[43]]-<span id='citeF-45'></span>[[#cite-45|45]]].
 ==ACKNOWLEDGEMENTS==

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 |}
-It is interesting to analyze the steady-state form of Eqs.(59) for the Stokes flow case where the convective terms are neglected. Now  the convective matrix '''A''' and the stabilization matrix '''L''' can be made equal to zero in the momentum equations and the resulting system can be written as
+It is interesting to analyze the steady-state form of Eqs.([[#eq-59|59]]) for the Stokes flow case where the convective terms are neglected. Now  the convective matrix <math>A</math> and the stabilization matrix <math>L</math> can be made equal to zero in the momentum equations and the resulting system can be written as
 <span id='eq-61'></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 |}
-The stability of the numerical solution is ensured by the presence of matrix '''L''' guaranteeing a positive definitiveness of the equation system ''for any choice of the approximations for v and'' <math display="inline">p</math>, thus overcoming the Babuska-Brezzi conditions <span id='citeF-1'></span>[[#cite-1|1]].
+The stability of the numerical solution is ensured by the presence of matrix <math>L</math> guaranteeing a positive definitiveness of the equation system ''for any choice of the approximations for v and'' <math display="inline">p</math>, thus overcoming the Babuska-Brezzi conditions <span id='citeF-1'></span>[[#cite-1|[1]]].
 The extension of above procedure to derive stabilized equations for incompressible solid mechanics problems is straight forward making use of the well known analogy between the Stokes equations for an incompressible flow and those of incompressible elasticity.
-Applications of the FIC method here proposed to incompressible problems in fluid and solid mechanics can be found in <span id='citeF-44'></span>[[#cite-44|44]].
+Applications of the FIC method here proposed to incompressible problems in fluid and solid mechanics can be found in <span id='citeF-44'></span>[[#cite-44|[44]],<span id='citeF-45'></span>[[#cite-45|45]]].
 ==9 CONCLUDING REMARKS==

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 where <math display="inline">N_j</math> are the linear shape functions, <math display="inline">n</math> is the number of nodes per element and <math display="inline">\overline{(\cdot )}</math> denotes nodal variables.
-Substituting Eqs.(58) into (57) leads to the following system of equations
+Substituting Eqs.([[#eq-58|58]]) into ([[#eq-57|57]]) leads to the following system of equations
 <span id='eq-59a'></span> <span id='eq-59b'></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"

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 |}
-Substituting Eqs.(55) into (53) and (54) gives after integration by parts of the deviatoric stress and pressure term of <math display="inline">r_{m_i}</math> in Eq.(53)
+Substituting Eqs.([[#eq-55|55]]) into ([[#eq-53|53]]) and ([[#eq-54|54]]) gives after integration by parts of the deviatoric stress and pressure term of <math display="inline">r_{m_i}</math> in Eq.([[#eq-53|53]])
 <span id='eq-57a'></span> <span id='eq-57b'></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"

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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>r_d - \underline{{1\over 2} h_{d_j} {\partial r_d \over \partial x_j}}{1\over 2} h_{d_j} {\partial r_d \over \partial x_j}=0  </math>
+| style="text-align: center;" | <math>r_d - \underline{{1\over 2} h_{d_j} {\partial r_d \over \partial x_j}}=0  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (44)